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From Quadratic Reciprocity to the Langlands Program

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From Quadratic Reciprocity to the Langlands Program From quadratic reciprocity to the Langlands program (Oliver Lorscheid, April 2018) In honor of the 2018 Abel prize winner Robert Langlands What is the Langlands program? The Langlands program consists of a web of conjectures, partially relying on conjectural objects, that connect L-functions for Galois representations and automorphic representations. A simple, but yet unproven and tantalizing instance is the following. Conjecture (Langlands 1970) For every positive integer n, there is a bijection 8 irreducible automorphic 9 irreducible representations < = Φ cuspidal representations GL −! ρ : LQ ! n(C) : π of GLn over Q ; such that if where is the L(ρ, s) = L(π; s) π = Φ(ρ) LQ (conjectural) Langlands group of Q, which is an extension of absolute Galois group Gal(Q=Q) of Q. Part 1: Reciprocity laws and class eld theory Quadratic reciprocity Let p be an odd prime number and a an integer that is not divisible by p. The Legendre symbol of a mod p is ( a 1 if a is a square modulo p, = p −1 if not. Conjecture (Euler 1783, Legendre 1785) Let and be distinct odd primes. Then p q Adrien-Marie Legendre q p p−1 q−1 = (−1) 2 2 : p q I First complete proof by Gauÿ in 1796. I Up to today, there are more than 240 dierent proofs. Carl Friedrich Gauÿ The p-adic numbers Let p be a prime number. The p-adic absolute value i a −i 0 is dened as whenever is not |·|p : Q ! R≥ p b p = p ab divisible by p. The p-adic numbers are the completion Qp of Q with respect to the norm . |·|p The -adic integers are 1 . p Zp = a 2 Qp jajp ≤ Since R is the completion of Q at the archimedean absolute value sign , we often write and to jaj1 = (a) · a Q1 = R p ≤ 1 express that p is a prime number or the symbol 1. Hilbert reciprocity Let a and b be rational numbers and p ≤ 1. The Hilbert symbol is (1 if 2 2 2 has a solution 3 ax + by = z (x; y; z) 2 Qp; (a; b)p = −1 if not. Theorem (Hilbert 1897) For all a; b 2 Q, we have (a; b)p = 1 for almost all p and Y (a; b)p = 1: p≤∞ David Hilbert Remark: Note that this implies quadratic reciprocity since for distinct odd prime numbers p and q, we have Y p−1 q−1 q p (p; q) = (p; q)2(p; q) (p; q) = (−1) 2 2 : ` p q p q `≤∞ Class eld theory and Artin reciprocity Theorem (Takagi 1920, Artin 1927) Let L=Qp be a nite Galois extension with abelian Galois group G = Gal(L=Qp) and norm N : L ! Qp, dened by Q . Then there is a N(a) = σ2G σ(a) group isomorphism Teiji Takagi × × ∼ Gal (−; L=Qp): Qp =N(L ) −! (L=Qp); which is called the local Artin symbol. Theorem (Artin 1927) Let L=Q be a Galois extension with abelian Galois Emil Artin group Gal ). Then Q 1 G = (L=Q p≤∞(a; L Qp=Qp) = × for all a 2 Q . Remark: This generalizes Hilbert's reciprocity law since p (a;L = )( b) p (a; b) = QppQp where L = [ b]. p b Q Chevalley's idelic formulation The idele group of Q is the group Q × × for almost all IQ = (ap) 2 p≤∞ Qp ap 2 Zp p < 1 : Note that × embeds diagonally into . Q IQ The idele class group of is ×. Q CQ = IQ=Q Theorem (Chevalley 1936) Claude Chevalley ab Let Q be the maximal abelian extension of Q. The product over all local Artin symbols denes a surjective group homomorphism Gal ab r : CQ −! (Q =Q) whose kernel is the connected component C 0 of the identity Q component of (w.r.t. the idelic topology on ). CQ CQ Part 2: L-series The Euler product In 1740, Euler describes explicit formulas for X 1 ζ(s) = ns n≥1 where s is an even positive integer. The case s = 2 solved the long standing Basel problem. Leonard Euler Moreover, Euler showed that Y 1 ζ(s) = 1 − p−s p<1 for real numbers s > 1, which is known as the Euler product nowadays. Dirichlet series × Let χ : Z ! C be a group homomorphism of nite order, i.e. χ(n) = 1 for some n ≥ 1. The Dirichlet series of χ is X χ(n) Y 1 L(χ, s) = = : ns 1 − χ(p)p−s n≥1 p<1 Application to primes in arithmetic progression: Peter Gustav Lejeune Theorem (Dirichlet 1840) Dirichlet Let a and n be positive integers. Then there are innitely many prime numbers p congruent to a modulo n. Riemann's analysis of ζ(s) Theorem (Riemann 1856) As a complex function, ζ(s) converges absolutely in the halfplane fs 2 CjRe(s) > 1g, has a meromorphic continuation to C with a simple pole at 1, and it satises a functional equation of the form Bernard Riemann ζ(1 − s) = (well-behaved factor) · ζ(s): In the following, we shall refer to such properties as arithmetic. In honor of Riemann's 1856 paper, ζ(s) is called the Riemann zeta function nowadays. Not to forget the Riemann hypothesis: Conjecture (Riemann 1856) If ζ(s) = 0, then s is an even negative integer or Re(s) = 1=2. Hecke L-functions Generalization of Dirichlet series (here only for Q): A Hecke character is a continuous group homo- morphism ×. It is unramied at χ : CQ ! C p if the composition × × is trivial. Zp ! CK ! C The Hecke L-function of χ is Y 1 L(χ, s) = : Erich Hecke 1 − χ(p)p−s χ unramied at p Theorem (Hecke 1916) L(χ, s) is arithmetic. If χ is nontrivial, then L(χ, s) is entire, i.e. without pole at 1. Problem: How to factorize large L-functions into smaller L-functions? Artin L-functions Let K=Q be a Galois extension with Galois group G = Gal(K=Q). An n-dimensional Galois representation of G is a continuous group homomorphism ρ : G −! GLn(C): If n = 1, then ρ is called a character. The Artin L-function of ρ is dened as the product Y L(ρ, s) = Lp(ρ, s) p<1 of local factors Lp(ρ, s), which are, roughly speaking, the reciprocals of the characteristic polynomials of ρ(Frobp) where Frobp is a lift of the Frobenius automorphism of the residue eld. Artin reciprocity, part 2 As a consequence of Artin's and Chevalley's theorems, the reciprocity map Gal ab induces an isomorphism r : CQ ! (Q =Q) ∗ Hom Gal ab × ∼ Hom 0 × r : ( (Q =Q); C ) −! (CQ=CQ; C ) between the respective character groups via r ∗(ρ) = ρ ◦ r. Theorem (Artin 1927) If χ = ρ ◦ r, then L(χ, s) = L(ρ, s). Applications: (1) Artin L-functions are arithmetic. (2) Factorization of large L-functions into smaller L-functions. The abelian Langlands correspondence Let Gal be the absolute Galois group of . GQ = (Q; Q) Q Since Gal( ab= ) = G ab and since every group homomorphism Q Q Q G ! × factors through G ab, we gain an isomorphism Q C Q Hom × Hom Gal ab × ∼ Hom 0 × (GQ; C ) = ( (Q =Q); C ) −! (CQ=CQ; C ): To extend this to an isomorphism with the whole character group of , we have to exchange by the Weil group of . CQ GQ WQ Q Theorem (Langlands correspondence for GL1) Artin reciprocity induces an isomorphism Hom × ∼ Hom × Φ: (WQ; C ) −! (CQ; C ) such that if . L(ρ, s) = L(χ, s) χ = Φ(ρ) André Weil Part 3: Going nonabelianthe Taniyama-Shimura-Weil conjecture Dirichlet series of a modular form The Poincaré upper half plane is H = fz 2 CjIm(z) > 0g. A modular cusp form (of weight k and level N) is an holomorphic function f : H ! C of the form Henri Poincaré 1 X 2 az + b f (z) = a e πinz=N s.t. f = (cz + d)k f (z) n cz + d n=1 for all a b SL congruent to 1 0 modulo . c d 2 2(Z) 0 1 N It is a (Hecke) eigenform if apq = apaq for all primes p 6= q. The Dirichlet series of f is X an Y 1 L(f ; s) = s = −s : n 1 − app n≥1 p<1 Theorem (Hecke 1936) L(f ; s) is arithmetic. Elliptic curves 1 1 An elliptic curve over Q is a Lie group E ' S × S (i.e. a complex torus) that is dened by equations over Q. In particular, it contains a subgroup on which acts. E(Q) ⊂ E GQ n n For a prime ` and n ≥ 1, let E(Q)[` ] be the subgroup of ` -torsion points of E(Q). The Tate module of E is T (E) = lim E( )[`n]; ` − Q n which is isomorphic to 2 and has an action of . Z` GQ For every good prime ` and embedding Z` ! C, we get a Galois representation John Tate GL GL2 ρ : GQ −! (T`(E) ⊗Z` C) ' (C): The L-function of E is the Artin L-function L(E; s) = L(ρ, s). The Taniyama-Shimura-Weil conjecture Conjecture (Taniyama 1955, Shimura 1957, Weil 1967) For every elliptic curve E over Q, there is a modular cusp eigenform f of weight 2 such that L(E; s) = L(f ; s). The conjecture was proven, step-by-step, by I Weil (1967), I Wiles (1995), with some help of Taylor, I Diamond (1996), Yukata Taniyama I Conrad, Diamond, Taylor (1999), I Brevil, Conrad, Diamond, Taylor (2001).
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